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Sensitivity of yield-per-recruit and spawning- biomass-per-recruit models to bias and imprecision in life history parameters: an example based on life history parameters of Japanese eel (Anguilla japonica)

Yu-Jia Lin1

Chi-Lu Sun (contact author)1

Yi-Jay Chang1

Wann-Nian Tzeng2

Email for contact author: chilu@ntu.edu.tw

1 Institute of Oceanography National Taiwan University No.1, Sec.4, Roosevelt Road Taipei 10617, Taiwan2 Department of Environmental Biology and Fisheries Science National Taiwan Ocean University No. 2, Beining Road Keelung 20224, Taiwan

Manuscript submitted 22 February 2014.Manuscript accepted 8 May 2015.Fish. Bull. 113:302–312 (2015).Online publication date: 2 June 2015.doi: 10.7755/FB.113.3.6

The views and opinions expressed or implied in this article are those of the author (or authors) and do not necessarily reflect the position of the National Marine Fisheries Service, NOAA.

Abstract—Yield-per-recruit and spawn ing-biomass-per-recruit mod-els, are commonly used for evaluat-ing the status of a fishery. In prac-tice, model parameters are them-selves usually estimates that are subject to both bias (uncertainty in the mean) and imprecision (uncer-tainty in the standard deviation). Using Monte Carlo simulation with data for female Japanese eel (An-guilla japonica) from the Kao-Ping River in Taiwan, we examined the sensitivity of such models to differ-ent degrees of bias and imprecision in the life history parameters. Posi-tive biases in natural mortality and the von Bertalanffy growth coeffi-cient led to larger relative changes in the mean and standard deviation of estimated fishing-mortality–based biological reference points (FBRPs) than did changes under negative biases. Higher degrees of impreci-sion in parameters did not affect the means of FBRPs, but their stan-dard deviations increased. Compos-ite risks of overfishing depended mainly on the changes in the means of FBRPs rather than on their stan-dard deviations. Therefore, reducing the biases in key life history para-meters, as well as the bias and im-precision in the current rate of fish-ing mortality, may be the most rel-evant approach for obtaining correct estimates of the risks of overfishing.

Yield-per-recruit (YPR) and spawn-ing-biomass-per-recruit (SPR) mod-els, in which the total yield or spawning biomass of a cohort is stan-dardized for the numbers of recruits, are commonly used in fisheries as-sessment (Beverton and Holt, 1957; Quinn and Deriso, 1999). They can be used to infer the total yield and spawning biomass of an entire popu-lation composed of different cohorts with an assumption of a steady state and with knowledge of equilibrium recruitment (King, 2007). Several fishing-mortality–based biological reference points (FBRPs) derived from YPR and SPR models can be used to evaluate whether the yield per re-cruit is optimal or the spawning bio-mass per recruit is sufficient for the population to persist under current fishing pressure.

Uncertainties in the parameters of such models are inevitable and result from observation and process error (Charles, 1998). Ignoring uncertain-ties in parameters can lead to incor-

rect estimation of FBRPs, and conse-quently the examination of fishery status could be misleading, given the cases of the American lobster (Homa-rus americanus) (Chen and Wilson, 2002), green sea urchin (Strongylo-centrotus droebachiensis) (Grabowski and Chen, 2004), Atlantic cod (Gadus morhua) (Jiao et al., 2005), prong-horn spiny lobster (Panulirus penicil-latus) (Chang et al., 2009), and Japa-nese eel (Anguilla japonica) (Lin et al., 2010a).

Estimation of natural mortality (M) is challenging (Vetter, 1988) be-cause both its mean and variance are prone to considerable uncertain-ty. For example, the estimates of M have varied among different empiri-cal methods (Pascual and Iribarne, 1993; Lin and Sun, 2013). The vari-ances of M estimates also have dif-fered among approaches (e.g., Cubil-los et al., 1999, versus Hall et al., 2004; Lin et al., 2012). It is easier to obtain growth information. The von Bertalanffy growth function is an

Lin et al.: Sensitivity of models to bias and imprecision in life history parameters 303

often used growth model, but different estimation ap-proaches (e.g., length-frequency methods versus read-ings of rings in calcified structures; see Lin and Tzeng, 2009, versus Lin and Tzeng, 2010), different regions of study (Helser and Lai, 2004; Keller et al., 2012), and aging errors from annuli readings (Lai and Gunderson, 1987; Cope and Punt, 2007) may lead to biases in the von Bertalanffy growth coefficient (K).

In another example of the potential for uncertainty, biases in the asymptotic length (L∞) may result from the regional variation in growth potential in different habitats (Beverton and Holt, 1957), from a latitudinal trend (Helser and Lai, 2004; Keller et al., 2012), or from sampling schemes unrepresentative of the popu-lation size structure, because of its high dependency of maximum size in the sample (Formacion et al., 1991; Froese and Binohlan, 2000). In addition, the uncertain-ty (or multiplicative error) in the growth curve (εGR) is related to the imprecision in the lengths-at-age scat-ter at a given age. The current fishing mortality rate (Fcur) can be estimated from various methods (e.g., Se-ber, 1982; Quinn and Deriso, 1999; King, 2007) with both the mean and variance of high uncertainty (Chen and Wilson, 2002).

It is essential to incorporate and quantify these un-certainties in the assessment of population dynamics (Hilborn and Walters, 1992; Peterman, 2004). Further, parameter uncertainties can be categorized into bias uncertainty and precision uncertainty. The influence of parameter bias on YPR and SPR models has been investigated since the 1950s (e.g., Beverton and Holt, 1957; Goodyear, 1993; Mace, 1994), but the effects of precision uncertainty have seldom been addressed. The effects of parameter imprecision with a limited number of comparisons (i.e., high or low variation) that were inadequate to reveal a full picture of the sensitivity to bias and imprecision in the parameters have been ex-amined in a few studies (e.g., Restrepo and Fox, 1988; Chen and Wilson, 2002; Chang et al., 2009). Systemat-ic examinations of different degrees of parameter bias and imprecision with a wider range and finer resolu-tion could provide detailed information about their ef-fects on model outputs (e.g., Goodyear, 1993), but few such examinations exist for per-recruit models.

It is difficult to elucidate the sensitivity of per-recruit models to parameter bias and imprecision through analysis of the formulae because the effects of the parameters on FBRPs are nonlinear with compli-cated forms (Schnute and Richard, 1998). Alternatively, a sensitivity analysis through the use of a Monte Carlo simulation can be used to examine their effects on the estimation of FBRPs, on fishery status, and on manage-ment implications (Jiao et al., 2005). Therefore, the ob-jective of this study was to evaluate the effects of dif-ferent degrees of bias, represented by the differences in parameter means, and of imprecision, represented by the differences in parameter standard deviations (SDs), in the YPR and SPR model results, specifically in the FBRPs and the resultant composite risks of overfishing.

Materials and methods

To better understand the sensitivity of outputs from the YPR and SPR models, we changed the mean or SD of selected parameters one at a time over large ranges with fine increments (from 5% to 95% with increments of 5% for cases of under-specification and from 150% to 1000% with increments of 50% for cases of over-specification). The following parameters were selected: 1) M; 2) von Bertalanffy growth coefficients of K and L∞; and 3) Fcur. In addition, the multiplicative error in the growth curve (εGR) and length–weight relationship (εLW) were selected as well. In this study, we used in-formation on the mean and SD values of these parame-ters from previous studies conducted during 1998–2006 on the life history and fishery of the Japanese eel in the Kao-Ping River in Taiwan (Lin and Tzeng 2009; Lin et al. 2010a, 2010b) to examine the sensitivity of model results to effects of parameter bias and impreci-sion with a Monte Carlo simulation.

We used wide ranges for the parameters for 3 rea-sons: 1) the scale of boundaries used; a factor of 10 was applied in a classical study (figure 4 in Goodyear, 1993); 2) a wide range is more general because it can include all likely ranges; and 3) in our previous study (Lin and Sun, 2013), we found that the variation in estimates of M from several different indirect methods can vary from 0.10/year to 2.13/year. Also, the difference in K can be large, especially when different approaches are involved. For the same region, for example, in Kao-Ping River, K has varied from 0.12/year to 0.38/year, depend-ing on whether methods based on otoliths or length frequencies were applied (Lin and Tzeng, 2009, 2010).

YPR and SPR models

The YPR and SPR models were calculated according to the formulae in Quinn and Deriso (1999):

YPR= e−M (tc−tr ) F(t)N(t)W(t)dt

tc

tmax∫ and (1)

SPR= N(t)W(t)S(t)dt

tc

tmax∫ , (2)

where F(t) = the fishing mortality at age t; N(t) = population size in numbers at age t; W(t) = the weight at age t; S(t) = the maturation proportion at age t; tr = the age at recruitment; tc = the age at first capture; and tmax = the maximum age (the formulae for F(t),

N(t), W(t), and S(t) are shown in Table 1).

Because, in Kao-Ping River, females were 4 times more abundant than males (Han and Tzeng, 2006) and be-cause females contribute more directly to spawning biomass, the parameters of female Japanese eels were used in this study; they were also used in Lin et al. (2010a). Estimates of means and corresponding SDs of the parameters were derived from our previous works (Lin and Tzeng, 2009; Lin et al.,2010b).

304 Fishery Bulletin 113(3)

Four FBRPs were calculated to compare with Fcur(0.120/year, Lin et al., 2010b): Fmax, the fishing mortality at which yield per recruit is at its maximum; F0.1, the fishing mortality at which the increase of yield per recruit is only 10% of the increase of yield per recruit when fishing mortality is zero (King, 2007); and F30% and F50%, the fishing mortality rates at which the SPR is 30% and 50% of the SPR when fishing mortality is zero (Goodyear, 1993).

The reference points Fmax and F0.1 can be regarded as boundaries in order to constrain harvesting below the level within which the fish population can produce maximum sustainable yield (United Nations, 1995), and they usually are used as limiting and precaution-ary indicators of growth overfishing, respectively; fish-ing mortality rates above Fmax indicate that fish are caught before they reach optimal size given the rate of natural mortality and their growth rate (Gabriel and Mace, 1999; Quinn and Deriso, 1999). Fishing mortali-ties at 30% and 50% of SPR, compared with SPR at F=0, are considered to be the limiting and precaution-

ary levels for anguillid eels (ICES1). Therefore, F30% and F50% were considered the limiting and precaution-ary reference points for recruitment overfishing, where the spawners are insufficient in numbers to produce enough offspring to replace themselves (Sissenwine and Shepherd, 1987).

Incorporation of uncertainties in parameters

We used Monte Carlo simulation, which has been wide-ly applied to other species (e.g., Chen and Wilson, 2002; Grabowski and Chen, 2004; Chang et al., 2009; Lin et al., 2010a) to incorporate the uncertainties of the pa-rameters into the models. In each run, random errors for natural mortality (εM) and for coefficients in the logistic maturation curve (εβ0 and εβ1), as well as εLW

1 ICES (International Council for the Exploration of the Sea). 2002. Report of the ICES/EIFAC Working Group on Eels. ICES Council Meeting (C.M.) Documents 2002/ACFM:03, 55 p. [Available at website.]

Table 1

Formulae, estimates of model parameters, and corresponding simulated random errors for the length–weight relationship (a and b are coefficients, and εLW=the corresponding multiplicative error), N=the population size in numbers; von Ber-talanffy growth function (K=the growth coefficient (/year), L∞=the asymptotic length (mm), t0=the theoretical age at length of zero (year), and εGR=the mul-tiplicative error of the growth curve), M=the natural mortality (/year); εM=the standard deviation of M; Fcur=current fishing mortality (/year); fishing process under gear selection (α0 and α1=selection curve coefficients), and maturation process of female Japanese eel (Anguilla japonica) in the lower reach of the Kao-Ping River in southern Taiwan (β0 and β1=the maturation curve coefficients, and εβ0 and εβ1=the standard error for β0 and β1, respectively). The means and standard deviations of the parameters are from previous studies conducted in 1998–2006 on life history parameters and the fishery of Japanese eel in the Kao-Ping River, Taiwan (Lin and Tzeng 2009, Lin et al. 2010a, 2010b).

Formula Parameter

Weight at age t, W(t)

a, b = 4.56 × 10−8, 3.55

εLW ~ N(0, 3.93 × 10−4)

L∞ ,K, t0 = 1024, 0.12, −0.69

εGR ~ N(0, 3.10 × 10−2)Population size in number at age t, N(t)

M = 0.18/year

εM ~ N(0, 1.98 × 10−5)Fishing process under logistic gear selection, F(t)

Fcur ~ Gamma with mean of

0.12 and variance of 1.06 × 10−2, α0, α1= −4, 0.02Maturation (silvering) proportion at age t, S(t)

β0, β1= −13.31, 0.02 εβ0 ~ N(0, 1.36) and εβ1 ~ N(0, 4.04 × 10

−6)Others

Maximum observed age, tmax 16 yearsLength and age at recruitment, Lr , tr 55 mm, 0.489 yearLength and age at first capture, Lc, tc 200 mm, 1.15 year

W(t)= aL(t)beeLW

L(t)= L∞[1− e−K (t−t0 ) ]eeGR

N(t)= e−[F (t)+(M+eM )](t−tr )

F(t)= Fcur×e[a0+a1L(t)]

1+ e[a0+a1L(t)]

S(t)=e[(b0+eb0)+(b1+eb1)L(t)]

1+ e[(b0+eb0)+(b1+eb1)L(t)]

http://www.ices.dk/sites/pub/Publication Reports/Expert Group Report/acfm/2001/wgeel/WGEEL01.pdf

Lin et al.: Sensitivity of models to bias and imprecision in life history parameters 305

and εGR, were generated from normal distribution with the means and variances listed in Table 1. These ran-dom errors were incorporated into the YPR and SPR formulae provided in Table 1, and then 4 FBRPs (Fmax, F0.1, F30%, and F50%) were calculated. For each simula-tion scenario, this process was repeated 5000 times to produce 5000 corresponding sets of FBRP values from which the empirical distributions of the 4 FBRPs were generated and composite risks were calculated.

Calculation of composite risks

Because Fcur and the FBRPs are not fixed constants, we applied composite risk analysis that allowed for the incorporation of the uncertainty in both indicator and management reference points (Prager et al., 2003; Jiao et al., 2005). By the discrete approach proposed by Jiao et al. (2005), the composite risks were calculated as the expected probability of one random variable being larger than another. Let f(x) be the empirical probabil-ity density function of the FBRP of interest (e.g., Fmax) and g(y) be the empirical probability density function of Fcur with a corresponding cumulative density func-tion of G(y) and then, let Dx be a small increase in x (i.e., the FBRP), and by summing x over its range, the composite risk of Fcur exceeding FBRP is calculated with the following equation:

P(Fcur > FBRP =1− [G(x) f (x)]−∞∞∑ Dx (3)

Here, we assume a gamma distribution for Fcur because 1) our previous study (Lin and Tzeng, 2008) indicated that gamma distribution fitted better for the distribu-tion of fishing efforts, 2) it produces nonnegative val-ues, and 3) it is flexible in shape. For details in cal-

culation of composite risks in discrete approach, refer to Jiao et al. (2005).

Sensitivity analysis

For the sensitivity analysis, the pa-rameter values in Table 1 were set as the standard scenario (scenario 1; Ta-ble 2). The parameters for which mean and SD values may potentially affect results of YPR and SPR models were investigated in 9 scenarios, in which the mean or SD of only one parameter was changed (Table 2): the mean of M (scenario 2) and its SD (εM, scenario 3), the mean of K (scenario 4), and the mean of L∞ (scenario 5). In practice, the bias in K can result from aging er-ror and the use of different estimation approaches, and the bias in L∞ often results from unrepresentative sam-pling schemes. Therefore, we assumed that the sources of biases in K and L∞ are unrelatedand they were modeled

independently. Because modeling the estimation errors in the coefficients K and L∞ can lead to underestima-tion of the actual uncertainty in the data (Lin et al., 2012), the error was modeled on the growth curve (εGR) rather than coefficients (scenario 6). The remaining scenarios are εLW (scenario 7) and the mean and SD values of Fcur and εFcur (scenarios 8 and 9).

For scenarios 1–7, the mean and SD values of the parameters, except for L∞, were decreased to 10% with an increment of 5% or were increased to 1000% with an increment of 50% to cover the possible magnitudes of biological variation (as applied in Goodyear, 1993). Because information about L∞ can be obtained from the maximum observed length in the data, that pa-rameter may be subject to less bias and imprecision and was set from 50% to 100% with an increment of 5% and from 100% to 200% with an increment of 10% (Table 2). For scenarios 8 and 9, because of less computational load (calculation of an FBRP is inde-pendent of Fcur, and, therefore, we needed to compute only composite risks of overfishing), we used finer in-crements: the mean and SD of Fcur were decreased to 10% with an increment of 1% or increased to 1000% with an increment of 10%.

The relative change (RC) between the mean or SD of one FBRP for increment i in scenario j (RCFBRP

ij ) and the mean or SD of that FBRP in the standard scenario was used to quantify the sensitivity of a given FBRP for all scenarios except 8 and 9. RC is defined with the following equation:

RCFBRP

ij =100×TFBRPij ×(TFBRP

S )−1,

(4)

Where TFBRPij

= the statistic (mean or SD) of the FBRP

of interest (e.g., Fmax) for increment i in scenario j; and

Table 2

Summary of the 9 scenarios for evaluation of the sensitivity of the results from models of yield per recruit and spawning biomass per recruit to dif-ferent degrees of bias and imprecision in several parameters: natural mor-tality (M), von Bertalanffy growth coefficient (K), asymptotic length (L∞), and current fishing mortality rate (Fcur). Also the multiplicative error in the growth curve (εGR) and the length–weight (LW) relationship (εLW) are modeled. In each scenario, the mean or standard deviation (SD) of only one parameter was changed.

Scenario Description Parameter Range

1 Standard scenario Unchanged 100%2 Mean of M M 10–1000%3 SD of M εM 10–1000%4 Mean of K K 10–1000%5 Mean of L∞ L∞ 50–200%6 Error in growth curve εGR 10–1000%7 Error in LW relationship εLW 10–1000%8 Mean of Fcur Fcur 10–1000%9 SD of Fcur εFcur 10–1000%

306 Fishery Bulletin 113(3)

TFBRP

S

= the same statistic of the same FBRP from

the standard scenario.

The sensitivity of an FBRP also has to be distinguished from the random variation that results from Monte Carlo simulation, namely the variation that results from the incorporation of random errors in M, length-weight relationship, and maturation process. To under-stand the scale of this random variation, the standard scenario (scenario 1, without any biasesor impreci-sion in parameters) was repeated 1000 times, produc-ing1000 corresponding RC values for the means and SDs of FBRPs. The minimum and maximum values of RC from these 1000 repetitions of scenario 1 were used as the lower and upper limits of random variation from the Monte Carlo simulation. The effect of changes in the mean or SD of a parameter was considered sig-nificant only when its RC is lower than the minimum or larger than the maximum RC. All the computations and simulations were completed in R, vers. 3.1.0 (R Core Team, 2014).

Results

Magnitude of the random variation from Monte Carlo simulation

Running the Monte Carlo simulation of the stan-dard scenario 1000 times produced RC values for the means of Fmax, F0.1, F30%,and F50% that ranged

from 99.95% to 100.04%, from 99.96% to 100.03, from 99.97% to 100.03, and from 99.99% to 100.03%, respectively (Table 3). The SD values of the FBRPs had larger RC values that ranged from 97.23% to 103.89% for Fmax, from 97.24% to 103.90% for F0.1, from 97.32% to 104.44% for F30%, and from 97.31% to 104.43% for F50%(Table 3). Therefore, variation of RC within 0.05% for the mean and within 4.50% for the SD of the FBRP was applied as the criterion for significance of ef-fects; that is, we considered the effect of changing one parameter on the FBRP to be significant when the resultant RC was wider than this criterion.

Sensitivity of reference points from the YPR model

The means of FBRPs from the YPR model, namely Fmax and F0.1, were sensitive to both the means of M and K, which had similar trends but different magnitudes (Fig. 1, left panels). The RC values for the means of Fmax and F0.1 decreased slowly to 75–80% when M and K decreased to 5%. As M and K increased to 1000%, their

RC values rose with an accelerating trend, especially for Fmax, to more than 12,000% and 1100%, respectively. However, the means of Fmax and F0.1 were not sensi-tive to changes in L∞ and εGR because their RC varied less than 0.05%. The εM had marginal effects on the mean of Fmax and F0.1, given that the RC was around 100.6% when εM became 1000% (Fig.1, left panels). The SD values of Fmax and F0.1 were also sensitive to the means of M and K with similar accelerating trends. The SD of Fmax was more sensitive to changes in M and K than was the SD of F0.1 (Fig. 1, right panels). In addi-tion, the SD values of Fmax and F0.1 were nearly related linearly to εM. Similar to the means, the SD values of Fmax and F0.1 did not show significant sensitivity to L∞ and εGR, and their RC values did not exceed the significance level of 4.50%.

Sensitivity of reference points from the SPR model

The means of FBRPs from the SPR model, F30%, and F50%, were sensitive to the mean of M, but their RC values (from 90% to 170%; Fig. 2, left panels) were smaller than those of Fmax and F0.1. When K decreased to 40%, the RC for the mean of F30% and F50% reached a minimum of around 80%. As K increased to 1000%, the RC of the means of F30% and F50% increased to 300% and 400%. They also were affected by L∞ such that the RC was around 85–125% as L∞ increased from 50% to 200%. Neither εM nor εGR affected the means of F30% and F50%. The SD values of F30% and F50% showed the highest sensitivity to the mean of M, with

Table 3

Mean and standard deviation (SD) per year of 4 fishery-mortality–based reference points(FBRPs)—the fishing mortality at which yield per recruit is at its maximum (Fmax),the fishing mortality at which the increase of yield per recruit is only 10% of the increase of yield per recruit when fishing mortality is zero(F0.1), and the fishing mortality rates at which the spawning biomass per recruit (SPR) is 30% and 50% of the SPR when fishing mortality is zero(F30%and F50%)—and the composite risks, P(Fcur>FBRP), calculated as percentages, of current fishing mortality, Fcur(0.120/year), exceeding FBRPs after repeating the standard scenario 1000 times. Numbers in the parentheses provide the ranges of relative changes in means and SDs from random Monte Carlo simulation with data for female Japanese eel (Anguilla japonica) in the Kao-Ping River in southern Taiwan.

FBRP Mean SD

Fmax 0.156 (99.95–100.04%) 1.6×10–3 (97.23–103.89%)F0.1 0.111 (99.96–100.03%) 9.2×10–4 (97.24–103.90%)F30% 0.129 (99.97–100.03%) 8.7×10–4 (97.32–104.44%)F50% 0.073 (99.97–100.03%) 4.6×10–4 (97.31–104.43%)

P(Fcur>Fmax) 13.44 0.47P(Fcur>F0.1) 56.44 0.70P(Fcur>F30%) 35.48 0.65P(Fcur>F50%) 94.31 0.32

Lin et al.: Sensitivity of models to bias and imprecision in life history parameters 307

Figure 1Relative changes determined from Monte Carlo simulation with data collected during 1998–2006 for female Japanese eel (Anguilla japonica) in the Kao-Ping River in southern Taiwan. The relative changes are measured as percentages for the means (shown in the left panels) and standard deviations (SD; shown in the right panels) of fishery-mortality–based refer-ence points(FBRPs) in scenarios 2–6 of the yield-per-recruit model, where the mean and SD of the natural mortality (M and εM), the von Bertalanffy growth coefficient (K), the asymptotic length (L∞), and the multiplicative error in the growth curve (εGR) were under-specified from 5% to 95% and over-specified from 150% to 1000% (except forvalues of L∞ that went from 50% to 200%). In each graph, the solid line indicates fishing mortality at which yield per recruit is at its maximum, and the dashed line indicates fishing mortality at which the increase of yield per recruit is only 10% of the increase of yield per recruit when fishing mortality is zero. Because of the large variation in RC values for the different scenarios, the scales of the y-axes differ greatly to allow changes to be seen. Under=underspecified parameters; Over=overspecified parameters; εM=the SD of M; Fcur=current fishing mortality and; εFcur=SD of Fcur.

Under Over Under Over

the RC changing from 70% to around 4000%. The pa-rameter K had high influence on the SD values of F30% and F50%, with RC changing from 20% to 1500% (Fig. 2, right panels). The parameters εM and L∞ had con-siderable effects on the SD values of F30% and F50%, with RC ranging from 90% to 400% and from 40% to 150%, respectively. The SDs of F30% and F50% were not affected by εGR.

Sensitivity of the composite risks of overfishing

The composite risks of growth overfishing (Fcur ex-ceeding Fmax and F0.1) decreased with increasing

means of M and K and did not depend on changes in L∞, εGR, and εM (Fig. 3). The risks of recruitment overfishing (Fcur exceeding F30% and F50%) showed the greatest sensitivity to K followed by M. The risk of recruitment overfishing was affected also by chang-es in L∞ but with less sensitivity. The parameters εM and εGR did not contribute to noticeable changes in the risks of overfishing. The risks of both growth and recruitment overfishing showed strong sensitivity to both the mean and SD of Fcur. As the mean of Fcur increased from around 150% to 225%, the risks ap-proached 100%. The risks of overfishing decreased with increasing SD for Fcur.

Rel

ativ

e ch

ang

e (%

)

Parameter magnifier (%) Parameter magnifier (%)

308 Fishery Bulletin 113(3)

Effects of multiplicative error in the length–weight relationship

The changes in εLW resulted in nonsignificant changes in the mean and SD of the 4 FBRPs. They also did not affect the composite risks of exceeding these FBRPs.

Discussion

Effects of uncertainty in mean and SD of natural mortality

Estimates of the mean of M are usually highly uncer-tain, possibly because of the lack of appropriate data

for direct estimation, data such as those from tag–re-capture experiments (Vetter, 1988). As M increases, the YPR curve becomes flatter with a decreasing maximum value and decreasing slope at the origin (fig. 17.18 in Beverton and Holt, 1957). This change in the shape of YPR curve accounts for the nonlinear accelerating trend of Fmax and F0.1 found in our study. Uncertainty in the mean (bias) of M can also cause significant bi-ases in results of the model for SPR, particularly at a high level of fishing mortality (Goodyear, 1993). A higher mean of M led to a steeper SPR curve, but the general shape of the curve was not altered, explain-ing the lower sensitivity of F30% and F50% to M. On the other hand, the SD (imprecision) of M resulted

Figure 2Relative changes from the Monte Carlo simulation with data collected during 1998–2006 for female Japanese eel (Anguilla japonica) in the Kao-Ping River in southern Taiwan. The relative changes are measured as percentages, for the means (shown in the left panels) and SDs (shown in the right panels) of fishery-mortality–based reference points(FBRPs) in sce-narios 2–6 of the spawning-biomass-per-recruit model, where the mean and SD of the natural mortality (M and εM), the von Bertalanffy growth coefficient (K), the asymptotic length (L∞), and the multiplicative error in the growth curve (εGR) were under-specified from 5% to 95% and over-specified from 150% to 1000% (except forvalues of L∞ that went from 50% to 200%). In each graph, the solid line indicates the fishing mortality rate at which the spawning biomass per recruit (SPR) is 30% of the SPR when fishing mortality is zero, and the dashed line indicates the fishing mortality rate at which the SPR is 50% of the SPR when fishing mortality is zero. Because of the large variation in RC values for the different scenarios, the scales of the y-axes differ greatly to allow changes to be seen. Under=underspecified parameters; Over=overspecified parameters.

Under Over Under Over

Rel

ativ

e ch

ang

e (%

)

Parameter magnifier (%) Parameter magnifier (%)

Lin et al.: Sensitivity of models to bias and imprecision in life history parameters 309

Figure 3The composite risks of overfishing in scenarios 2–6, 8, and 9, where the mean and standard deviation (SD) of the natural mortality (M and εM), the von Bertalanffy growth coefficient (K), the asymptotic length (L∞), the multiplicative error in the growth curve (εGR), and the mean and SD of current fishing mortality (Fcur and εFcur) were under-specified from 5% to 95% and over-specified from 150% to 1000% (except forvalues of L∞ that went from 50% to 200%). The black solid line indicates the composite risk of Fcur exceeding the fishing mortality at which yield per recruit is at its maximum, the black dashed line indicates the composite risk of Fcur exceeding the fishing mortality at which the increase of yield per recruit is only 10% of the increase of yield per recruit F=0,and the gray solid and dashed lines indicate the composite risk of Fcur exceeding the fishing mortality rates at which the spawning biomass per re-cruit (SPR) is 30% and 50% of the SPR when F=0.

in significant changes only in the SDs of the 4 FBRPs examined. Therefore, as indicated in the Appendix Figure, the risks of both growth and recruitment overfishing were affected more by the changes in the mean of Mthan by changes in the SD. The bias in M seems to be more im-portant than the imprecision in M in influencing the risks of overfishing.

Effects of uncertainty in growth parameters

The shape of the YPR curve also was altered by K (fig. 17.22 in Beverton and Holt, 1957). The flattening of the YPR curve with decreas-ing K accounts for the accelerating increase in the mean and SD of Fmax. On the other hand, Beverton and Holt (1957) found that the changes in W∞(equivalent to L∞ under the same length–weight relationship) did not affect the shape of the YPR curve. This insensitivity of the YPR shape on L∞ explains our finding that changes in L∞ did not affect the mean and SD of Fmax and F0.1 or the corresponding risk of growth overfishing.

The peaks in the values of RC in the means and SDs of F30% and F50% when K and L∞were around 40–60% possibly resulted from the use of a length-dependent maturation curve for an-guillids (Davey and Jellyman, 2003) in the cal-culation of SPR. In the cases with extremely small values of K or L∞, the proportion of ma-ture eels was close to zero even at the maxi-mum age (16 years). A very small part of the spawning biomass was lost because of fishing, consequently resulting in higher F30% and F50% values.

Effects of current fishing mortality

The bias and imprecision in Fcur played an im-portant role in determining risks of overfishing. Greater effects due to changes in the mean of Fcur on the risks of both growth and recruit-ment overfishing were expected because differ-ences in the mean played an important part in influencing the composite risk, as in the ex-ample of 2 standard normal random variables (Appdx Fig.). Given the same difference in the means, a larger SD leads to lower composite risks (Appendix. Figure, panels B and D), ac-counting for the observed decreasing risks of overfishing with increasing εFcur. Given that Fcur is from the gamma distribution, decreas-ing composite risks with increasing εFcur also resulted in the convergence of 4 risks of over-fishing. This finding indicates that a high εFcur value may mask the distinction between target (usually F0.1 or F50%) and threshold (Fmax or F30%) FBRPs because the risks of target and threshold Fbrp are similar.

Under Over

310 Fishery Bulletin 113(3)

The assumption of a gamma distribution of Fcur may be the reason for increasing risks of exceeding Fmax when εFcur lies between 5% and 200%. When εFcur was 50% or less, the gamma distribution of Fcur resembled the normal distribution with density highly concentrat-ed around its mean. Because Fmax (0.151) was much larger than Fcur (0.120), the area of the distribution of Fcur that exceeded Fmax became smaller and smaller as εFcur decreased, resulting in lower risks.

Asymmetry in the sensitivities of reference points

The responses of the means and SDs of 4 FBRPs to the changes in the means of M or K showed disproportion-ate increases, indicating that the means and SDs of FBRPs were more sensitive to over-specification. This accelerating trend, arising from the nonlinear relation-ship between these parameters and FBRPs (Schnute and Richards, 1998), indicates that under the same degree of misspecification (e.g., 50%) an over-specified mean of M or K could result in seriously overestimated values of FBRPs. Consequently, the risks of overfish-ing would be underestimated, potentially leading to overexploitation.

In summary, bias in life history parameters, such as M, Fcur, K, and L∞, resulted in considerable changes in the means and SDs of 4 selected FBRPs: Fmax, F0.1, F30%, and F50%. Different degrees of the imprecision in the life history parameters did not affect the means of FBRPs but substantially influenced their SDs. Over-specification of the mean of M and K led to larger val-ues of RC in the means and SDs of FBRPs than did under-specification of the means of M and K.

The composite risks of Fcur exceeding these 4 FBRPs were affected mainly by bias in the life history pa-rameters rather than by their imprecision. Both bias and imprecision in Fcur played crucial roles in deter-mining the risks of Fcur exceeding these 4 FBRPs. The variation in growth curves and length–weight rela-tionships did not affect results of the per-recruit mod-els. Our results agreed with those of previous studies without consideration of parameter uncertainty, indi-cating that they can apply to YPR and SPR models for other species. We recommend 1) minimizing the bias in the life history parameters, especially bias in M, K, and Fcur, and 2) maximizing the precision in Fcur as the most relevant approaches for development of correct estimates of FBRPs and determining risks of overfishing.

Acknowledgments

This study is financially supported by the Aims to Top University Project, National Taiwan University. We are thankful to A. Punt, School of Aquatic and Fish-ery Sciences, University of Washington, for providing comments on the early draft, B. Jessop, Department of Fisheries and Oceans, Bedford Institute of Oceanogra-phy, for providing both scientific and grammatical com-

ments on the manuscript, and to anonymous reviewers for their suggestions.

Literature cited

Beverton, R. J. H., and S. J. Holt.1957. On the dynamics of exploited fish populations, 533

p. Great Britain Fish. Investig. Ser. II. Ministry of Agriculture, Fisheries and Food, Fisheries Laboratory, Lowestoft, U.K.

Chang, Y. J., C.-L. Sun, Y. Chen, S.-Z. Yeh, and W.-C. Chiang. 2009. Incorporating uncertainty into the estimation of

biological reference points for a spiny lobster (Panu-liruspenicillatus) fishery. N.Z.J. Mar. Freshw. Res. 43:429–442. Article

Charles, A. T. 1998. Living with uncertainty in fisheries: analytical meth-

ods, management priorities and the Canadian ground fishery experience. Fish. Res. 37:37–50. Article

Chen, Y., and C. Wilson.2002. A simulation study to evaluate impacts of uncer-

tainty on the assessment of American lobster fish-ery in the Gulf of Maine. Can. J. Fish. Aquat. Sci. 59:1394–1403. Article

Cope, J. M., and A. E. Punt.2007. Admitting ageing error when fitting growth curves:

an example using the von Bertalanffy growth func-tion with random effects. Can. J. Fish. Aquat. Sci. 64:205–218. Article

Cubillos, L. A., R. Alarcón, and A. Brante.1999. Empirical estimates of natural mortality for the

Chilean hake (Merlucciusgayi): evaluation of preci-sion. Fish. Res. 42:147–153.

Davey, A. J. H., and D. J. Jellyman.2003. Sex determination in freshwater eels and manage-

ment options for manipulation of sex. Rev. Fish Biol. Fish. 15:37–52. Article

Formacion, S. P., J. M. Rongo, and V. C. Sambilay Jr.1991. Extreme value theory applied to the statistical dis-

tribution of the largest lengths of fish. Asian Fish. Sci. 4:123–135. [Available at website.]

Froese, R., and C. Binohlan.2000. Empirical relationships to estimate asymptotic

length, length at first maturity and length at maxi-mum yield per recruit in fishes, with a simple meth-od to evaluate length frequency data. J. Fish Biol. 56:758–773. Article

Gabriel, W. L., and P. M. Mace.1999. A review of biological reference points in the context

of the precautionary approach. In Proceedings of the fifth national NMFS stock assessment workshop: pro-viding scientific advice to implement the precautionary approach under the Magnuson-Stevens Fishery Conser-vation and Management Act; Key Largo, FL, 24–26 Feb-ruary 1998 (V. R. Restrepo, ed.), p. 34–45. NOAA Tech. Memo. NMFS–F/SPO–40.

Goodyear, C. P. 1993. Spawning stock biomass per recruit in fisheries

management: foundation and current use. In Risk evaluation and biological reference points for fisheries management (S. J. Smith, J. J. Hunt, and D. Rivard, eds.), p. 67–81. Can. Spec. Publ. Fish. Aquat. Sci. 120.

http://dx.doi.org/10.1080/00288330909510012http://dx.doi.org/10.1016/S0165-7836(98)00125-8http://dx.doi.org/10.1139/f02-102http://dx.doi.org/10.1139/f06-179http://dx.doi.org/10.1007/s11160-005-7431-xhttp://www.asianfisheriessociety.org/publication/abstract.php?id=760http://dx.doi.org/10.1111/j.1095-8649.2000.tb00870.x

Lin et al.: Sensitivity of models to bias and imprecision in life history parameters 311

Grabowski, R., and Y. Chen.2004. Incorporating uncertainty into the estimation of the

biological reference points F0.1 and Fmax for the Maine green sea urchin (Strongylocentrotus droebachiensis) fishery. Fish. Res. 68:367–371. Article

Hall, H. G., S. A. Hesp, and I. C. Potter.2004. A Bayesian approach for overcoming inconsisten-

cies in mortality estimates using, as an example, data for Acanthopagrus latus. Can. J. Fish. Aquat. Sci. 61:1202–1211. Article

Han, Y.-S., and W.-N. Tzeng.2006. Use of the sex ratio as a means of resource as-

sessment for the Japanese eel Anguilla japonica: a case study in the Kaoping River, Taiwan. Zool. Stud. 45:255–263.

Helser, T. E., and H.-L. Lai.2004. A Bayesian hierarchical meta-analysis of fish

growth: with an example for North American large-mouth bass, Micropterus salmoides. Ecol. Model. 178:339–416. Article

Hilborn, R., and C. J. Walters1992. Quantitative fisheries stock assessment: choice,

dynamics and uncertainty, 570 p. Chapman & Hall, London.

Jiao, Y., Y. Chen, and J. Wroblewski.2005. An application of the composite risk assessment

method in assessing fisheries stock status. Fish. Res. 72:173–183. Article

Keller, A. A., K. J. Molton, A.C. Hicks, M. Haltuch, and C. Wetzel.

2012. Variation in age and growth of greenstriped rockfish (Sebastes elongatus) along the U.S. west coast (Washing-ton to California). Fish. Res. 119:80–88. Article

King, M.2007. Fisheries biology, assessment and management, 2nd

ed., 400 p. Wiley-Blackwell, Oxford, U.K.Lai, H. L., and D. R. Gunderson.

1987. Effects of ageing errors on estimates of growth, mortality and yield per recruit for walleye pollock (Theragra chalcogramma). Fish. Res. 5:287–302.

Lin, Y.-J., S.-L. Chang, M.-Y. Chang, S.-H. Lin, T.-I. Chen, M.-S. Su, W.-C. Su, and W.-N. Tzeng.

2010b. Comparison of recapture rates and estimates of fishing and natural mortality rates of Japanese Eel An-guilla japonica between different origins and marking methods in a mark-recapture experiment in the Kaop-ing River, southern Taiwan. Zool. Stud. 49:616–624.

Lin, Y.-J., Y.-J. Chang, C.-L. Sun, and W. N. Tzeng.2010a. Evaluation of the Japanese eel fishery in the

lower reaches of the Kao-Ping River, southwest-ern Taiwan using a per-recruit analysis. Fish. Res. 106:329–336. Article

Lin, Y.-J., N.-J. Su, B. M. Jessop, W.-C. Chiang, and C.-L. Sun.2012. Errors in coefficient or expected value? Effects of

different methods on simulated values for a linear mod-el and the nonlinear von Bertalanffy growth model. J. Fish. Soc. Taiwan 39:23–34.

Lin, Y.-J., and C.-L. Sun.2013. Estimation of natural mortality of Japanese eel

(Anguilla japonica) using multiple indirect methods. J. Fish. Soc. Taiwan 40:171–182.

Lin, Y.-J., and W.-N. Tzeng.2008. Effects of shrimp net and cultured eels on the wild

population of Japanese eel Anguilla japonica in the Kao-Ping River, Taiwan. J. Fish. Soc. Taiwan 35:65–73.

2009. Modelling the growth of Japanese eel Anguilla ja-ponica in the lower reach of the Kao-Ping River, south-ern Taiwan: an information theory approach. J. Fish Biol. 75:100–112. Article

2010. Vital population statistics based on length frequen-cy analysis of the exploited Japanese eel (Anguilla ja-ponica) stock in the Kao-Ping River, southern Taiwan. J. Appl. Ichthyol. 26:424–431. Article

Mace, P. M.1994. Relationships between common biological refer-

ence points used as thresholds and targets of fisher-ies management strategies. Can. J. Fish. Aquat. Sci. 51:110–122. Article

Pascual, M. A., and O. O. Iribarne.1993. How good are empirical predictions of natural mor-

tality? Fish. Res. 16:17–24.Peterman, R. M.

2004. Possible solutions to some challenges facing fish-eries scientists and manager. ICES J. Mar. Sci. 61:1331–1343. Article

Prager, M. H., C. E. Porch, K. W. Shertzer, and J. F. Caddy.2003. Target and limits for management of fisheries: a

simple probability-based approach. N. Am. J. Fish. Manage. 23:349–361. Article

Quinn, J. T., II, and R. B. Deriso.1999. Quantitative fish dynamics, 560 p. Oxford Univ.

Press, New York.R Core Team.

2014. R: a language and environment for statistical com-puting. R Foundation for Statistical Computing, Vien-na, Austria. [Available at website, accessed May 2014.]

Restrepo, V. R., and W. W. Fox Jr.1988. Parameter uncertainty and simple yield-per-recruit

analysis. Trans. Am. Fish. Soc. 117:282–289. ArticleSeber, G. A. F.

1982. The estimation of animal abundance and related parameters, 2nd ed., 672 p. Charles Griffin & Co., London.

Schnute, J. T., and L. J. Richards.1998. Analytical models for fishery reference points. Can.

J. Fish. Aquat. Sci. 55:515–528. ArticleSissenwine, M. P., and J. G. Shepherd.

1987. An alternative perspective on recruitment over-fishing and biological reference points. Can. J. Fish. Aquat. Sci. 44: 913–918. Article

United Nations.1995. Agreement for the implementation of the provisions

of the United Nations Convention on the Law of the Sea of 10 December 1982 relating to the conservation and management of straddling fish stock and highly migratory fish stocks. United Nations Conference on Straddling Fish Stocks and Highly Migratory Fish Stocks, Sixth session, New York, 24 July–4 August 1995. A/CONF. 164/37, 40 p. [Available at website.]

Vetter, E. F.1988. Estimation of natural mortality in fish stocks: a re-

view. Fish. Bull. 86:25–43.

http://dx.doi.org/10.1016/j.fishres.2003.12.005http://dx.doi.org/10.1139/f04-074http://dx.doi.org/10.1016/j.ecolmodel.2004.02.013http://dx.doi.org/10.1016/j.fishres.2004.11.003http://dx.doi.org/10.1016/j.fishres.2011.12.012http://dx.doi.org/10.1016/j.fishres.2010.08.015http://dx.doi.org/10.1111/j.1095-8649.2009.02268.xhttp://dx.doi.org/10.1111/j.1439-0426.2010.01453.xhttp://dx.doi.org/10.1139/f94-013http://dx.doi.org/10.1016/j.icesjms.2004.08.017http://dx.doi.org/10.1577/1548-8675(2003)023%3C0349:TALFMO%3E2.0.CO;2http://www.r-project.org/http://dx.doi.org/10.1577/1548-8659(1988)117%3C0282:PUASYA%3E2.3.CO;2http://dx.doi.org/10.1139/f97-212http://dx.doi.org/10.1139/f87-110http://www.un.org/depts/los/fish_stocks_conference/fish_stocks_conference.htm

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Appendix FigureThe composite risks of Y exceeding X, or P(Y>X), measured in percent-ages, in which X is a standard normal variable (X~N[0,1]) and Y is a random normal variable with changing mean and standard deviation (SD) (μY, σY). (A) Reference case: (μY, σY)=(from 0 to 1, from 0.05 to 1); (B) large SD of Y case: (μY, σY)=(from 0 to 1, from 0.5 to 10); (C) large difference in mean case: (μY, σY)=(from 0 to 10, from 0.05 to 1); and (D) large difference in mean and SD of Y case: (μY, σY)=(from 0 to 10, from 0.5 to 10).

Appendix

This appendix presents changes in composite risks of 2 random variables, X and Y, under different means and standard deviations.

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